I read Tom Braden's paper on mixed category for toric varieties. At least one idea is useful, namely where does the extra grading come from.

The idea is that, given a vector space with a unipotent operator, then you can split the vector space into Jordan blocks. Each Jordan block has a Z-indexed-filtration, with the index well-defined upto a global shift.

Suppose you have T^*P^n, with the stratification given by P^0, P^1, ... , like in category O. Then we want to equip the wrapped Fukaya category's cocore's endomorphism with extra grading.

Combinatorially, we can try to match the $End(T)$ algebra with the combinatorially defined $A(V)$ algebra. $A(V)$ algebra has extra grading. You can say that this is like $T$-equivariant cohomology, where the generator has cohomologically grading $2$. Then, we forget the cohomological grading and only keep the extra grading. Then the question is, why the endomorphism of $T$ brane is the same as $A(V)$. You can say that Gammage-Mcbreen-Webster solved the problem, but not quite. Local system is naturally identified with coherent sheaves module over torus; Only unipotent local system is identified with module near the identity of the torus; and only unipotent system with an explicit Frobenius action allows you to have a $\Z$-index filtration. So, the Fukaya category needs a graded lift.

There should be a version of the A-side, whose endomorphism of co-core is isomorphic to $A(V)$ as an ungraded algebra, then we can choose some graded lift. Well, the crossing over (between strands of different color) should have grading $1$, dot having grading $2$. That's the grading convention of $A(V)$.

Now, what's the relation between

• Frobenius on a variety $X$ defined over $\F_q$. we get, mixed constructible sheaf.
• algebraic Lagrangian skeleton in a algebraic symplectic variety, over $\F_q$. we should get mixed microlocal sheaf.
• holomorphic Lagrangian skeleton in a holomorphic symplectic variety, over $\C$. we should get mixed microlocal sheaf.
• Ask Saito, what's the analog in $\C$ for the mixed constructible sheaf?

In principle, all these question should be answered by these Koszul duality people. Koszul duality is about, two holomorphic skeletons, and a duality correspondence that sends cocore to core. It has been proven in the realization of mixed DQ module; but not yet in the relatization of Fukaya category.

First of all, Reeb chord, is not like Morse critical point. So, we don't necessarily have a canonical grading on it. It depends on a choice of holomorphic symplectic structure. There is some Conley-Zehnder index that I don't quite know. Maybe only $\Z/2$-grading is canonical. But that is Morse grading.

Let's dream a bit. Consider the 3d mirror between $T^*\P^2$ and $A_2$ surface, the smoothing of $\Z^2/\mu_3$ (we say $A_1$ surface is the smooth $T^*\P^1$).

What's the classical story of Koszul duality? Quadratic duality.

• symmetric algebra vs. anti-symmetric algebra.
• symmetric algebra but with a bit more quadratic relations; less anti-symmetric algebra. I don't have a geometric intuition for what does that mean. I don't understand why quadratic plays a big role here.

There is another way to think about this. Say, you have some semi-simple ring $R_0 = \oplus k e_\alpha$, and then some graded ring $R = R_0 \oplus R_1 \cdots $. So we have some 'augmentation', $R \to R_0$, most likely sending all the $R_{>0} \to 0$. Then, we should have some $R^! = End_{R-mod}(R_0)$. That might serve as the dual algebra. maybe graded? diagonally graded?

OK fine.

How about the algebra $A(V)$, it is positively graded, with degree $0$ part semi-simple. When the conormal intersects with the 'diagonal Lagrangian' (ok, I don't really know what does that mean? what is this diagonal Lagrangian in $T^*\R^n$? There is no good looking one. We can say

Suppose we have a region $\Delta_\alpha \In \eta + V$ and dual region $\Delta^\vee_\alpha \In V^\vee_\xi$, They don't necessarily intersect. Projectively, they might intersect.

Consider the following construction. Starting from $T^*\C^N$, with Hamiltonian $(\C^*)^N$-action, with alg moment map to $\C^N$ with singularity hyperplanes. OK, take a slice $\eta + V_\C \In \C^N$ (taking complex moment map), then take a quotient.

Consider SES $$ V_\Z \to \Z^N \to (V^\perp)^*_\Z $$ and $$ V^\perp \to (\Z^N)^* \to V^*_\Z $$ We want to say that $(V^\perp_\Z)^* \otimes \C$ is like the dual Lie algebra $Lie(L)^*$ of $L \In T=(\C^*)^N$. Then, $Lie(T)^* \cong \Z^N$.

• We are alg symp reduction by $L = (V^\perp_\Z)\otimes \C^*$ to get $M_H(V)$.
• We are alg symp reduction by $(V_\Z)\otimes \C^*$ to get $M_H(V)$.

It is sort of wrong to say $(V_\Z)\otimes \C^*$ and $(V^\perp_\Z)\otimes \C^*$ are two different subgroups of the same torus. They are different. Although the dual torus of $\C^*$ is $\C^*$, it is not the same one. So, we should not say that they are reduction from the same torus.

However, maybe we can take the product $M_H(V) \times M_H(V^\vee)$. We can say that, we have $T^*(\C^{2N})$, and do Hamiltonian reduction by a Lagrangian torus $V_{\C^*} \times V^\perp_{\C^*}$, and is left with still a product space.

Take the Coulomb branch and Higgs branch point of view. The additive Coulomb branch is given by the BFN construction. The quantized additive Coulomb branch is some homology on BFN space, which is related to $H_* Map(S^2, N/G)$. The Higgs branch is $T^*(N/G)$. The Coulomb branch is like based loop in some manifold, and the Higgs branch is like cohomology of that space itself.

(what about category $O$? what about the two sets of parameters, Kahler and equivariant. )

What did Ben Webster do?